[EM] Tideman and GMC

[Steve Eppley]

The following is part of a message which has been languishing 
for a couple of months, not entirely completed, in my outgoing 
message queue:

Markus S wrote:
> Dear Steve, you wrote (23 Feb 2000):
> > Blake C wrote:
-snip-
> > > So, my conclusion is that Tideman (winning-votes) was rejected
> > > on faulty grounds.
> >
> > I agree with Blake.  The so-called "Tideman bad example" 
> > (which Mike Ossipoff had posted) showed Tideman defeating an 
> > alternative which lost only one pairing, and that pairing was 
> > 1 to 0, and that outcome looked dubious.  Schulze would have 
> > elected that alternative.
> 
> So you disagree with Mike Ossipoff's Tideman bad example.
> But do you agree with my Tideman bad example (3 Feb 2000)?

No, I don't consider Markus' Feb 3 example to be a bad example 
for Tideman either.  For reference, here's the Feb 3 example:

   26 voters vote:  C>A>B>D
   20 voters vote:  B>D>A>C
   18 voters vote:  A>D>C>B
   14 voters vote:  C>B>A>D
    8 voters vote:  B>D>C>A
    7 voters vote:  D>A>C>B
    7 voters vote:  B>D>(A=C)
       ==>  Majorities:  BD75, CB65, DC60, AD58, AB51, CA48

   Tideman locks BD75, locks CB65, skips DC60 (because DC would
   cycle with the locked majorities), locks AD58, locks AB51, 
   and locks CA48.  The locked majorities imply that the 
   "best ranking" is C>A>B>D, so Tideman elects C.

   Schulze compares strongest opposing beatpaths:
   ADCB58 > BDCA48     implies A finishes ahead of B.
   ADC58  > CA48       implies A finishes ahead of C.
   AD58   > DCA48      implies A finishes ahead of D.
   CB65   > BDC60      implies C finishes ahead of B.
   BD75   > DCB60      implies B finishes ahead of D.
   CBD65  > DC60       implies C finishes ahead of D.
   Schulze elects A.  (The implicit Schulze ranking is A>C>B>D,
   though I'm not sure if Markus will acknowledge this.)

   (For the record, BCM is indecisive between A&C.  IBCM,
   after eliminating all but A&C, elects C due to CA48.)

Markus claimed, if I haven't misinterpreted him, that the 
example shows Tideman is more manipulable than Schulze, by 
pointing out that some of the voters who prefer B to D and A to 
C (the 20 "B>D>A>C" voters) can change the Tideman outcome from 
C to A by not ranking B ahead of D (since that would reduce the 
BD majority to 59 or smaller so DC60 would get locked in).

What Markus apparently neglected to notice is that voters can 
easily manipulate Schulze in this scenario too, to change the 
Schulze winner from A to C.  Some of the voters who prefer C 
to A and A to D (the 26 "C>A>B>D" voters and the 14 "C>B>A>D" 
voters) can uprank D ahead of A or downrank A behind D, which 
would reverse the A>D majority to D>A.  Since the ADC58 beatpath 
depends on the A>D majority, the ADC58 beatpath would be 
destroyed and Schulze would elect C.

Here's a second consideration, but less important:  Markus' 
Tideman strategizers (some of the 20 "B>D>A>C" voters) need to 
suppress the expression that B is their favorite -- which we may 
be consider a drastic or near-drastic strategy -- in order to 
improve the outcome from their least preferred alternative to 
their second least preferred alternative.  By contrast, the 
Schulze strategizers would not have to suppress the expression 
of their favorite, and would change the outcome to their 
favorite, and a smaller percentage of the relevant faction(s) 
need to strategize.

So it seems to me we should not accept the argument that the 
example shows Tideman(majoritarian) to be more manipulable 
than Schulze.  (Tideman(margins) is more manipulable, though.)


---Steve     (Steve Eppley    seppley at alumni.caltech.edu)